Re: Pi = 4

When Archimedes did his calculuations he used a lower bound as well. Then he used the squeeze theorem to say that the value of Pi must be inbetween the two values. The above diagram only gives you an upper bound on the value of Pi.

Re: Pi = 4

Originally Posted by TheEmptySet

When Archimedes did his calculuations he used a lower bound as well. Then he used the squeeze theorem to say that the value of Pi must be inbetween the two values. The above diagram only gives you an upper bound on the value of Pi.

Can't you have a similar polygonal path whose length is arbitrarily close to 4 inside the circle?

Re: Pi = 4

Originally Posted by emakarov

Can't you have a similar polygonal path whose length is arbitrarily close to 4 inside the circle?

I do not think so. With a little bit of calculus you should be able to convice yourself that the side length of an inscribed square with maximum periemter would have length So the starting Perimeter would be

Re: Pi = 4

For a similar fallacy, let the broken line from (0, 0) to (1/2, 0) to (1/2, 1/2) to (1, 1/2) to (1, 1) be an "approximation" to the straight line from (0, 0) to (1, 1)- it has length 1/2+ 1/2+ 1/2= 2. A better approximation would be the broken line from (0, 0) to (1/4, 0) to (1/4, 1/4) to (1/2, 1/4) to (1/2, 1/2) to (3/4, 1/2) to (3/4, 3/4) to (1, 3/4) to (1, 1) for a total length of 8(1/4)= 2 again. In general, you have 2n line segments each of length 1/n. "In the limit" this appears to converge to the straight line, which has length , but the total length is always 2.